Transcript Rotation Torque, Rolling, & Angular Momentum

Rotation
Torque, Rolling, & Angular Momentum
(Chapters 10 & 11, p.241-304)
Rotation: Terminology
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Axis of rotation- the line/axis around which an object spins
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r – the radius of a spinning thing
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 (theta) – a variable for the angle that something has rotated through
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s – a variable for the linear distance (arc length) that an object at radius “r” would spin through
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 - angular velocity (the angle-that-is-rotated-through per unit-of-time: radians/second)
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 - angular acceleration (change in angular-velocity per unit-of-time: radians/second^2)
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v – “linear” (or “tangential”) velocity
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T - period (the time needed for object to rotate back to “start angle”)
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K – as always, the “Kinetic energy”
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I – rotational inertia (somewhat analogous to typical, linear inertia)
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T - torque (a sort of “rotational force”)
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W – work
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P – power
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l – angular momentum
The Basics of Rotation
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1 rev=360=2π radians
Instantaneous angular velocity is derivative of function
for angular position: (t)
Angular acceleration is the derivative of velocity: (t)
The same acceleration-a/displacement-d/velocity-v
equations (from kinematics) are used for //
s (the arc length) = r
…taking the derivative of above: v=r, and a=r
Using centripetal equations (acentripetal / radial=v2/r) and
v=r, we get:
aradial=2r
Energy in Rotation
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Kinetic Energy= ½I2
Every rotating particle has an “I” equal to its-mass times
its-radius-from-the-axis
…thus a big object’s “I” is the integral of its particles:
I=∫r2dm
(all the infinitesimally small masses, times
each’s radius-from-axis)
d
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If you know “I” for an object when it spins around a
certain axis that goes through the center-of-mass, BUT it
is spinning around a new, different axis (that happens to
be parallel to the old one)…
Inew=Iorig+(mobject)(distance-between-axes)
Rotational Forces (Torque), Work,
and Power
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Net Torque = I
(sort of like F=ma)
Just as work=F*Δx=∫Fdx,
Work = TΔ = i∫f Td = ΔK
Power=dW = T
dt
Rollin’, Rollin’, Rollin’!
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If a wheel spins through a certain angle, the distance it moves on
the ground equals the arc-length that corresponds to that angle… (if
you get that good, if you don’t, forget it, it’s ok)
…thus, “distance-across-ground”=“arc-length”:
“d”=“s”=r
…taking the derivative of both sides, we get:
v=r
(v=linear velocity along ground)
…deriving again:
a=r
Note that d, v, and a refer to the motion of the rolling-object’s centerof-mass.
Energy:
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Kinetic Energy=Krotational+Klinear= ½I2 + ½mv2
Torque Related to Linear Force
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The torque (or the-”rotational force”) being
applied to a particle equals the cross product of
that-particle’s-position-vector (r) and the-forceon-the-particle (a vector called “F”):
T=rxF
…by simply using the definition of a cross
product, we get: T=|r|*|F|*sinΦ
…the basic point is that T equals the product of
the PERPENDICULAR-COMPONENTS-of-r-and-F
Angular Momentum
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l =“angular-momentum”= rxp = m(rxv)
(…where “p” is linearmomentum)
…for just l’s magnitude: l =|r|*m*|v|*sinΦ, where Φ is the anglebetween-vector-r-and-vector-p
For a system of particles, the system’s-total-angular-momentum
(called “L”) is just the sum of each particle’s-angular-momentum:
L= l 1+ l 2+ l 3+…= Σl
For a rigid body, the component of its angular-momentum that is
parallel to its axis-of-rotation is found by:
Lparallel=I
The total angular momentum is conserved; that is: L-initial=L-final
Question 1
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What would be the axis of rotation for a
car wheel?…for a spinning quarter?
The axle; a line that is the diameter of the
quarter’s face and that cuts Washington’s
face into two parts
Question 2
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While an object’s linear position is its xcoordinate (or coordinates), its angular
position is __________.
The angle that it is turned to.
Question 3
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Angular displacement is _____.
The angle through which the object has
been rotated…. Or “the final angularposition minus the initial angular-position”
Question 4
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(a) A wheel spins one full revolution every
second. (b) That spinning causes that
wheel’s center of mass (and the car it is
attached to) to travel 4 feet per second
along the road.
These are two types of
velocities; what is each called?
A>>angular velocity;
B>>linear velocity
Question 5
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D
You mark a point on a wheel’s edge. The point
is at an angle theta at time t, such that
(t)=3t^3 + t^2 + t
…which of the
following is a function describing the angular
acceleration of the point?
A) alpha=0
B) alpha=9t^2 + 2t
C) alpha=9t^2 +2t +1 D) alpha=18t +2
Question 6
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If roll of toilet paper rolls along the ground
and rotates an angle equal to “theta,” how
much paper will roll off? (how far along
the ground will it roll?)
Length-of-paper-to-roll-off = arc-lengththat-corresponds-to-theta-radians =
radius-times-theta
Question 7
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Linear velocity equals relates to angular
velocity in what way?
Linear-velocity = angular-velocity * radius
Question 8
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How does Period relate to angular
velocity? And how does it relate to linear
velocity?
T = (2pi)/
T=(2pi*radius)/v
Question 9
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What is the rotational inertia of a cylinder,
rolling on it’s side?
I=.5Mr^2 where r is radius and M is mass
Question 10
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What is the rotational inertia of a rod spun
like a baton?
I=(1/12)*ML^2, where M is the mass and L
is the length
Question 11
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What is the rotational inertia of a tire
shaped object, rotating as a tire does on a
car wheel?
I=.5M*(a^2 + b^2), where a is the inner
radius, and b is the outer radius
Question 12
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What is the main equation of Newton’s
Second Law as applied to rotation?
Torque=I*alpha
Question 13
Force-Vector

Radius-Vector
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The force above is applied to a wheel’s edge,
but is not applied tangential to that edge. How
is the torque-applied calculated?
Torque-applied = |r|*|F|*sin (this is the same as
multiplying the magnitude-of-the-radius and themagnitude-of-the-component-of-F-that-is-perpendicularto-the-radius)
Question 14
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Work is the integral of what function? (possible
hint: when you integrate, you find the area of a
function; in a way, you are sort of multiplying
the x-values of the function by the y-values… so
whatever two things multiply to equal “Work,”
those are the x and y of your function)
Y
X
Work is the integral of Force as a function of
distance (or, “x”)
Question 15
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Power is the integral of what function?
What is the independent variable of this
function?
Work as a function of time
Question 16
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What is the the parallel axis theorem?
If the rotational inertia is calculated for an object
spinning around a certain axis that goes through
the center-of-mass, but then a new, different
axis-of-rotation is used (and the new axis is
parallel to the old axis, but does not go through
the center of mass), then:
Inew= Icenter-of-mass+Mass*distance-between-axes
Question 17
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If a wheel rolls smoothly, the distance it
will move (linearly, across the ground) will
equal the ___-length that corresponds to
the angle the wheel rotated through.
The arc-length
Question 18
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A rolling wheel experiences which kind of
friction?
A)static or B)kinetic
A) Static
(because the wheel never
actually moves, or “skids” along the
surface… if it were perfect it would simply
touch down on the ground, then lift up as
it rolled)
Question 19
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What is one way to calculate angularmomentum (l )? (give an equation)
It equals “the cross-product of a radiusvector and a linear-momentum-vector”. It
also equals “mass multiplied by the crossproduct of radius and linear-velocity”.
Question 20
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If you take the derivative of angularmomentum (with respect to time), what
do you get? (possible hint: what do you
get when you take the derivative of linearmomentum as a function of time?)
You get the net-torque
Problem 1
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A wheel (of radius=25cm) makes 2.5 revolutions
every second as it rolls smoothly. What is it’s
linear velocity? How long will it take to roll a
linear distance of “3.25*pi” meters? And what is
it’s rotational period?
1.25*pi (meters per second)
2.5 seconds
.4 seconds
Problem 2
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At any time “t” (in seconds) a rotating
object is at an angle “theta” (in radians)
and Theta= t^4 + 4t^3 +2t^2 +5. What
is the instantaneous angular acceleration
at second-2? What is the average angular
acceleration between seconds 1 and 3?
100 radians per second-squared
187 radians per second-squared
Problem 3
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Through what angle will an object rotate
given that: it starts at an angular-velocity
of pi radians-per-second, it has a constant
angular acceleration of 3*pi radians-persecond-squared, and it goes for 2 seconds
? Be sure not to over-think this one.
8*pi radians
Problem 4
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At the bottom of a ramp, a smoothly
rolling tire is traveling at an angular speed
of 3 rad-per-sec. The inner radius of the
tire is 50cm; the outer is 1m. It’s mass is
12kg. How high will the rolling tire go up
the ramp?
0.066 meters
h?
Problem 5
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Force applied
Gary
30°
Radius-vector
Gary is on a miniature, “spinable” merry-go-round on the
play ground, standing .5 meters from the center. Larry
then applies a force of 11 Newtons to the edge of the
merry-go-round to spin Gary. The edge is 1.5 meters in
radius from the center. BUT, the force is applied not
tangentially (at an angle of 90-degrees from the radiusvector), but at an angle of 30-degrees to the radiusvector. If Gary and the merry-go-round altogether
masses at 300kg, then how fast, linearly, will Gary
accelerate?
.014 m/s^2
Problem 6
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Force= 4^2 (where =0 radians just as Larry starts)
What will be the overall change in energy of the merrygo-round-AND-Gary system from problem #5, if the
above equation gives the Force applied by Larry (still
applied at 30-degrees) as a function of the merry-goround’s angular position “theta” AND IF Larry spins it
two full revolutions?
+64*pi^3 Joules
Problem 7
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A boy in a marching band is smoothly spinning a .75-meter-long
baton of uniformly distributed mass about an axis that is
perpendicular to the baton and goes through it’s center. It spins at
3*pi rad-per-sec. He lets go of it quickly, without getting in its way
or altering its rotation. He then quickly grabs the end of it and spins
the baton about a new axis that is parallel to the old one, but that
goes through the rod’s end. The baton quickly ends up having a
new angular speed (with respect to the new axis) of 1*pi rad-persec. The baton has a mass of 0.3kg. What is the minimal amount of
work the rod did upon the band-member’s hand? (ignoring gravity
and air resistance…)
0.35 Joules
Problem 8
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At one instant, force-vector “F=4.0ĵ N” acts on a
0.25 kg object that has position-vector “r=(2.0î2.0k) meters” and velocity-vector “v=(5.0î+5.0k) m/s”. About the origin and in unitvector notation, what are (a) the object’s
angular momentum and (b) the torque acting on
the object?
0
(8.0 Newton-meters)î + (8.0 Newton-meters)k
Problem 9
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A wheel is rotating freely at an angular speed of
800 rev/min on a shaft whose rotational inertia
is negligible. A second wheel, initially at rest
and with twice the rotational inertia of the first,
is suddenly coupled to the same shaft. (a) What
is the angular speed of the resultant
combination of the shaft and two wheels? (b)
What fraction of the original rotational kinetic
energy is lost?
a) 267 rev/min
b) 0.0667
Problem 10
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Two 2.00 kg balls are attached to the ends of a thin rod of length
50.0 cm and negligible mass. The rod is free to rotate in a vertical
plane without friction about a horizontal axis through its center.
With the rod initially horizontal (see figure), a 50.0 g wad of wet
putty drops onto one of the balls, hitting it with a speed of 3.00 m/s
and then sticking to it. (a) What is the angular speed of the system
just after the putty wad hits? (b) What is the ratio of the kinetic
energy of the system after the collision to that of the putty wad just
before? (c) Through what angle will the system rotate before it
momentarily stops?
putty
a) 0.148 rad/s
b) 0.0123
c) 181 degrees
Axis of
rotation
Closing Note
Always remember the
difference between a
positive and negative
torque. It is not that
one is “up” and one’s
“down.” Rather,
remember that they
are “clockwise” and
“counterclockwise.”
Negative
torque
(that
happens to
point
down)
Positive
torque
(that
happens
to point
down)